In this paper, based on Bochner formula, mass decay estimates and elliptic Moser iteration, we show the global existence and asymptotic convergence of a subsequence of solutions of Calabi flow on some closed 3-manifolds, and then the existence of extermal metrics of $L^{2}$-norm of scalar curvature functional on a fixed conformal class is claimed. In particular, we may re-solve part of the Yamabe conjecture on closed 3-manifolds.

We consider quotient surfaces by $p$-closed rational vector fields. First we show that singularities on the quotient surfaces by multiplicative $p$-closed vector fields are tonic singularities. Then we proceed to studying global properties of Zariski surfaces. We see that non-trivial global 1-forms are related to some linear systems with base points. We also give examples of Zariski surfaces admitting non-closed regular 1-forms.

In this paper, we consider an arithmetic Hodge index theorem for a family of semi-stable curves, generalizing Faltings-Hriljac's arithmetic Hodge index theorem for an arithmetic surface.

We consider the spectral gap of the two demensional Ising model on a special graph with some special boundary conditions. The special graph is a finite square with a square hole on its center part, that is, we consider a finite square of side $2L +1$, and we remove another smaller finite square of side $2L_{1} +1(L_{1} <L)$ which has the same center as the finite square of side $2L +1$, therefore there is a square hole at the center part of the finite square of side $2L +1$, we denote this special graph by $\Lambda (L,L_{1})$. On this graph, the boundary of $\Lambda (L,L_{1})$ is composed of “inner boundary” and “outer boundary”. We will discuss two different boundary conditions of the Ising model on $\Lambda (L,L_{1})$, one is that the outer boundary condition is plus and minus spins’ “mixed” boundary condition and the inner boundary condition is plus boundary condition; the other is that the outer boundary condition is plus boundary condition and the inner boundary condition is an arbitrary boundary condition. On above two different boundary conditions, in the absence of an external field and at large inverse temperature $\beta$ we will show the upper bound of the spectral gap of Ising model for the first of above boundary conditions, and the lower bound of the spectral gap of Ising model for the second of above boundary conditions. These two results show that if we consider the first of above boundary conditions, and exchange this inner boundary condition with the outer boundary condition of the Ising model on $\Lambda (L,L_{1})$, the spectral gap of Ising model will be greately changed. The results can be extended to some other cases, for example, we can consider some other boundary conditions and some other graphs.